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COMP211 Computer Logic Design. Lecture 1. Number Systems. Prof. Taeweon Suh Computer Science Education Korea University. A Desktop Computer System ( as of 2008). The term “Processor” is used to refer to CPU. CPU. Main Memory (DDR2). FSB (Front-Side Bus). North Bridge. Graphics card.

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## Lecture 1. Number Systems

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**COMP211 Computer Logic Design**Lecture 1. Number Systems Prof. Taeweon Suh Computer Science Education Korea University**A Desktop Computer System (as of 2008)**• The term “Processor” is used to refer to CPU CPU Main Memory (DDR2) FSB (Front-Side Bus) North Bridge Graphics card DMI (Direct Media I/F) Peripheral devices South Bridge Hard disk USB PCIe card**A Computer System**• Computer is composed of many components • CPU (Intel’s Core 2 Duo, AMD’s Opteron etc) • Memory (DDR3) • Chipsets (North Bridge and South Bridge) • Power Supply • Peripheral devices such as Graphics card and Wireless card • Monitor • Keyboard/mouse • etc**Digital vs Analog**Analog Digital music video wireless signal**Bottom layer of a Computer**• Each component inside a computer is basically made based on analog and digital circuits • Analog • Continuous signal • Digital • Only knows 1 and 0**What you mean by 0 or 1 in Digital Circuit?**• In fact, everything in this world is analog • For example, sound, light, electric signals are all analog since they are continuous in time • Digital circuit is a special case of analog circuit • Power supply provides power to the computer system • Power supply has several outlets (such as 3.3V, 5V, and 12V)**What you mean by 0 or 1 in Digital Circuit?**• Digital circuit treats a signal above a certain level as “1” and a signal below a certain level as “0” • Different components in a computer have different voltage requirements • CPU (Core 2 Duo): 1.325 V • Chipsets: 1.45 V • Peripheral devices: 3.3V, 1.5V Note: Voltage requirements change as the technology advances 1.325V “1” Not determined “0” 0V time**Number Systems**• Analog information (video, sound etc) is converted to a digital format for processing • Computer processes information in digital • Since digital knows “1” and “0”, we use different number systems in computer • Binary and Hexadecimal numbers**Number Systems - Decimal**• Decimal numbers • Most natural to human because we have ten fingers (?) and/or because we are used to it (?) • Each column of a decimal number has 10x the weight of the previous column • Decimal number has 10 as its base ex) 537410 = 5 x 103 + 3 x 102 + 7 x 101 + 4 x 100 • N-digit number represents one of 10N possibilities ex) 3-digit number represents one of 1000 possibilities: 0 ~ 999**Number Systems - Binary**• Binary numbers • Bit represents one of 2 values: 0 or 1 • Each column of a binary number has 2x the weight of the previous column • Binary number has 2 as its base ex) 101102 = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 0 x 20 = 2210 • N-bit binary number represents one of 2N possibilities ex) 3-bit binary number represents one of 8 possibilities: 0 ~ 7**Power of 2**• 20 = • 21 = • 22 = • 23 = • 24 = • 25 = • 26 = • 27 = • 28 = • 29 = • 210 = • 211 = • 212 = • 213 = • 214 = • 215 =**Power of 2**• 20 = 1 • 21 = 2 • 22 = 4 • 23 = 8 • 24 = 16 • 25 = 32 • 26 = 64 • 27 = 128 * Handy to memorize up to 29 • 28 = 256 • 29 = 512 • 210 = 1024 • 211 = 2048 • 212 = 4096 • 213 = 8192 • 214 = 16384 • 215 = 32768**Number Systems - Hexadecimal**• Hexadecimalnumbers • Writing long binary numbers is tedious and error-prone • We group 4 bits to form a hexadecimal (hex) • A hex represents one of 16 values • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F • Each column of a hex number has 16x the weight of the previous column • Hexadecimal number has 16 as its base ex) 2ED16 = 2 x 162 + E (14) x 161 + D (13) x 160 = 74910 • N-hexadigit number represents one of 16N possibilities ex) 2-hexadigit number represents one of 162 possibilities: 0 ~ 255**Number Conversions**• Hexadecimal to binary conversion: • Convert 4AF16 (also written 0x4AF) to binary number • Hexadecimal to decimal conversion: • Convert 0x4AF to decimal number • 0100 1010 11112 • 4×162 + A (10)×161 + F (15)×160 = 119910**Bits, Bytes, Nibbles**• Bits (b) • Bytes & Nibbles • Byte (B) = 8 bits • Used everyday • Nibble (N) = 4 bits • Not commonly used**KB, MB, GB …**• In computer, the basic unit is byte (B) • And, we use KB, MB, GB many manymany times • 210 = 1024 = • 220 = 1024 x 1024 = • 230 = 1024 x 1024 x 1024 = • How about these? • 240 = • 250 = • 260 = • 270 = • … 1KB (kilobyte) 1MB (megabyte) 1GB (gigabyte) 1TB (terabyte) 1PB (petabyte) 1EB (exabyte) 1ZB (zettabyte)**Quick Checks**• 222 =? • 22 × 220 = 4 Mega • How many different values can a 32-bit variable represent? • 22 × 230 = 4 Giga • Suppose that you have 2GB main memory in your computer. How many bits you need to address (cover) 2GB? • 21 × 230 = 2 GB, so 31 bits**Addition**• Decimal • Binary**Binary Addition Examples**• Add the following 4-bit binary numbers 0001 1110**Overflow**• Digital systems operate on a fixed number of bits • Addition overflows when the result is too big to fit in the available number of bits • Example: • add 13 and 5 using 4-bit numbers**Signed Binary Numbers**• How does the computer represent positive and negative integer numbers? • There are 2 ways • Sign/Magnitude Numbers • Two’s Complement Numbers**Sign/Magnitude Numbers**• 1 sign bit, N-1 magnitude bits • Sign bit is the most significant (left-most) bit • Negative number: sign bit = 1 • Positive number: sign bit = 0 • Example: 4-bit representations of ± 5: +5 = 01012 - 5 = 11012 • Range of an N-bit sign/magnitude number: [-(2N-1-1), 2N-1-1]**Sign/Magnitude Numbers**• Problems • Addition doesn’t work naturally • Example: 5 + (-5) 0101 + 1101 10010 • Two representations of 0 (±0) 0000 (+0) 1000 (-0)**Two’s Complement Numbers**• Ok, so what’s a solution to these problems? • 2’s complement numbers! • Don’t have the same problems as sign/magnitude numbers • Addition works fine • Single representation for 0 • So, hardware designer likes it and uses 2’s complement number system when designing adders (inside CPU)**Two’s Complement Numbers**• Same as unsigned binary numbers, but the most significant bit (MSB) has value of -2N-1 • Example • Biggest positive 4-bit number: 01112 (710) • Lowest negative 4-bit number: 10002 (-23 = -810) • The most significant bit still indicates the sign • If MSB == 1, a negative number • If MSB == 0, a positive number • Range of an N-bit two’s complement number [-2N-1, 2N-1-1]**How to Make 2’s Complement Numbers?**• Reversing the sign of a two’s complement number • Method: • Flip (Invert) the bits • Add 1 • Example -7: 2’s complement number of +7 0111 (+7) 1000 (flip all the bits) + 1 (add 1) 1001 (-7)**Two’s Complement Examples**• Take the two’s complement of 01102 1001 (flip all the bits) + 1 (add 1) 1010 • Take the two’s complement of 11012 0010 (flip all the bits) + 1 (add 1) 0011**Two’s Complement Addition**• Add 6 + (-6) using two’s complement numbers • Add -2 + 3 using two’s complement numbers**Increasing Bit Width**• Sometimes, you need to increase the bit width when you design a computer • For example, read a 8-bit data from main memory and store it to a 32-bit • A value can be extended from N bits to M bits (where M > N) by using: • Sign-extension • Zero-extension**Sign-Extension**• Sign bit is copied into most significant bits. • Number value remains the same • Examples • 4-bit representation of 3 = 0011 • 8-bit sign-extended value: • 4-bit representation of -5 = 1011 • 8-bit sign-extended value: 00000011 11111011**Zero-Extension**• Zeros are copied into most significant bits. • Number value may change. • Examples • 4-bit value = 0011 • 8-bit zero-extended value: • 4-bit value = 1011 • 8-bit zero-extended value: 00000011 00001011**Number System Comparison**For example, 4-bit representation:

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